Innovative Teaching Exchange

by Su Liang, California State University San Bernardino

Having taught Intermediate
Algebra in University of Connecticut for three semesters, I found that a quite
number of students kept making a common mistake:

.

Even in a Calculus class,
this mistake is not uncommon in students’ work. Whenever they have an algebraic
expression containing like terms in the numerator and denominator, they simply
cancel them out. I had been thinking about how to correct the misconception
effectively. In a pedagogy class, we were assigned a project to find a common
misconception and to design some effective activities so that the misconception
can be corrected by letting students make sense out of it. I shared this problem
with two other mathematics teachers in my group. We decided to use this problem
as our project. Luckily, when I had a chance to teach Intermediate Algebra in
the fall 2008, I modified our design and applied it to my teaching. The result
is significant for my class. In the beginning of the semester, about 1/3 of the
students had the misconception. In the end of the semester, no students repeated
this type of mistake. I would like to share our approach for this problem.

To tackle a problem, we have
to know the underlying reason. Why are students tempted to make this type of
mistake?

In the beginning of the
semester, I let the students form their own groups to do group projects together
through the whole semester. Each group had three or four students. In total we
had 7 groups, 25 students. I held a group discussion and raised this question:
Is

,

Given that bâ? c and neither a
nor b is 0? I asked the groups to discuss this question and to reach an answer
with full explanation. Each group would have a representative present their
ideas. After about a 10-minute discussion, I asked the two groups who gave the
’yesâ? answers to give their reasons. They reasoned that because ’a’ is in the
numerator and also in the denominator, by division, it can be cancelled out.
Then I asked the groups who said ’noâ? to explain their answer. They gave the
reason that ’a’ can’t be cancelled because the operation sign between ’a’ and
’c’ and ’a’ and ’b’ are addition, not multiplication. Students could not wait to
hear my judgment. I told them that I would not tell them who is correct just
yet. By giving them the following work sheet, I let the groups work on it and
report their result when they finished.

Please use the given values for a, b, c to
calculate ,,ac, ab, ,
and .
Fill the results in the table.

a

b

c

1

2

9

2

3

1

3

5

4

4

6

3

5

1

2

6

2

4

7

4

3

8

3

1

9

1

2

After 15 minutes, all the groups finished their work.
I asked the groups one by one to report what they found. They all reported
that

in all
the cases of the table. I then asked them to make a conclusion based on the
data.

I didn’t stop there. Instead,
I raised another question: Can we prove our conclusion algebraically?
Again, I let the groups work on their own. I gave a hint to the groups who did
not have a clue what to do (Hint: using cross multiply). Ten minutes later, all
groups explained their ideas of proof. Following their explanations, I wrote
down the proof step by step on the board. First, we suppose that

.

Second, by
cross
multiplying
both sides, we
get
b(a + c) = c(a + b).
After distributing both sides, we have ab + bc = ac + bc.
Then we subtract bc from both sides, and we get ab = ac.
Dividing both sides by ’a’
(since a â? 0),
we reach the answer b = c,
which contradicts our conditionb â? c.
So clearly

is only
true algebraically if b = c.
Therefore,

.

That means that we can’t
cancel out ’a’ when given
.

After finishing the algebraic
proof, I asked the following questions: What is the difference between the two
algebraic expressions: and
?
Immediately, students answered that in the first one the relationships between
’a’ and ’c’ and ’a’ and ’b’ are addition but in the second one the
relationships between ’a’ and ’c’ and ’a’ and ’b’ are multiplication.
Then I asked them: based on
the operation rule, what is the order of operation for the first expression? ’We
do additions first and then divideâ?, some students answered. One student
responded: ’Aha, now I see why I can’t cancel out ’a’. Now it makes sense to
me.â? ’How about
I continued. Some students replied: ’based on operation rule, when
we do division and multiplication, we can do either operation first by the
operation ruleâ? ’Then
what can we do about ’a’ in this case?â? I asked further. A number of students
spoke out: ’’a’ can be cancelled by divisionâ?.

I addressed a summary to my
students: ’As you can see, according to the data from the work sheet of table,
the algebraic proof, and the operation rule, they all give us the same
conclusion:

.â?

I used a red chalk to
write this in a big size of letters on the board. Furthermore, I asked the
groups to write a report reasoning why
based
on what we did in the class.

Finally I gave them two
questions and let them write the answers:

1. Is (2x + 1)/(2x +2) =
1/2 ?

2. Is 2x/3x = 2/3 ?

I checked their answers one
by one. Every student in the class replied correctly. Through the end of this
semester, students did not repeat the type of mistake

It is worth
mentioning that although my work focused on a specific example, the method -
having students work out numerical examples, then discussing the algebra, then
looking at the differences between the situations where what the student wants
to do is CORRECT versus where it is not, and finally practicing with some more
sophisticated examples - is a promising one for other common student errors as
well.

Acknowledgment

I would to thank my classmates Carla Ryall and Christopher Dailey who worked with me for the group
project ’ finding a common misconception and designing some effective
activities so that the misconception can be corrected by letting students make
sense out of it in our pedagogy class in spring, 2008.

Su Liang
(liangs@csusb.edu)
graduated
from China University of Politics Science and Law with a B.A. in Business Law
and from University of Connecticut with a M.S. in mathematics. She is completing
her PhD. in Mathematics Education at University of Connecticut in summer 2010
and will be an assistant professor in California State University at San
Bernardino in fall 2010. Her research interest is K-12 mathematics teacher
preparation.